I am self-studying branched coverings. I read it from B. Maskit's Kleinian groups book. I want some more references for reading branched covers. In particular, I want to understand how to create branched covers of a given topological space or more precisely branched covers of surfaces.
If anyone shears related references like books or articles it will be a great help.
Montesinos wrote several papers defining the meaning of branched coverings and proving basic properties(not just between manifolds, but for general topological spaces):
Montesinos-Amilibia, José María, Branched folded coverings and 3-manifolds, Castrillón López, Marco (ed.) et al., Contribuciones matemáticas en honor a Juan Tarrés. Madrid: Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas (ISBN 978-84-695-4421-1). 295-315 (2012). ZBL1297.57008.
Despite the title, he works in much greater generality than 3-manifolds. In the paper he also refers to his earlier work (which I cannot access):
Montesinos-Amilibia, José María, Branched coverings after Fox, Bol. Soc. Mat. Mex., III. Ser. 11, No. 1, 19-64 (2005). ZBL1104.57002.
You should be able to get it through your interlibrary loan (I am not sufficiently motivated to do so myself).
Here is another paper, which you can find online and has basic definitions and properties. One advantage is that the paper also has a definition of a PL branched covering, which is much easier to digest:
Montesinos-Amilibia, José-María, Open 3-manifolds and branched coverings: a quick exposition, Rev. Colomb. Mat. 41, No. 2, 287-302 (2007). ZBL1149.57300.
Everybody else appear to be happy to work with a much more limited definition which I explain here.
Note, however, that there is a disagreement even about the meaning of the word "covering" which many traditionalists among complex geometers interpret simply as a locally biholomorphic map, something a topologist would find abhorrent.
You can have a look at Makoto Namba's book "Branched coverings and algebraic functions".
